Research Methodology - Dr. Krishan K. Pandey

Sampling Fundamentals 171
08 .
= ×
64
08 .
= ×
64
2400 − 64
2400 − 1
2336
2399
= (0.1) (.97)
= .097
(3) 90 per cent confidence interval for the mean number of accidents per intersection per year
is as follows:
R
S|
T|
σ s X ± z ×
n
N − n
N − 1
b gb g
= 3. 2 ± 1645 . . 097
U
V|
W|
= 32 . ± . 16 accidents per intersection.
When the sample size happens to be a large one or when the population standard deviation is
known, we use normal distribution for detemining confidence intervals for population mean as stated
above. But how to handle estimation problem when population standard deviation is not known and
the sample size is small (i.e., when n < 30 )? In such a situation, normal distribution is not appropriate,
but we can use t-distribution for our purpose. While using t-distribution, we assume that population is
normal or approximately normal. There is a different t-distribution for each of the possible degrees of
freedom. When we use t-distribution for estimating a population mean, we work out the degrees of
freedom as equal to n – 1, where n means the size of the sample and then can look for cirtical value
of ‘t’ in the t-distribution table for appropriate degrees of freedom at a given level of significance. Let
us illustrate this by taking an example.
Illustration 3
The foreman of ABC mining company has estimated the average quantity of iron ore extracted to be
36.8 tons per shift and the sample standard deviation to be 2.8 tons per shift, based upon a random
selection of 4 shifts. Construct a 90 per cent confidence interval around this estimate.
Solution: As the standard deviation of population is not known and the size of the sample is small, we
shall use t-distribution for finding the required confidence interval about the population mean. The
given information can be written as under:
X = 368 . tons per shift
σ s = 28 . tons per shift
n = 4
degrees of freedom = n – 1 = 4 – 1 = 3 and the critical value of ‘t’ for 90 per cent confidence interval
or at 10 per cent level of significance is 2.353 for 3 d.f. (as per the table of t-distribution).